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2. To clarify the cut-off procedures we consider the relation between the occupation number representation and the tensor product of Hilbert spaces in an axiomatic manner. Assumptions with respect to the states of a single particle are the following: S. The states of a single particle correspond to the vectors of a Hilbert space H. S. The physical quantities of a single particle correspond to the self-adjoint operators (defined on a dense subset of the space H). Sa. There is a set of physical quantities whose corresponding operators commute each other and makes a complete system of operators. We denote these operators by 0,..., 0. S. We restrict our considerations to the case that eigenvalues a,(a,,..., a,) of the operators (0,..., 0) makes a point (discrete) spectrum. Assumptions with respect to the states of quantized field are the following: F. To every system of eigenvalues ,(,,’’’,,n) of single particle, i.e. to every eigenstate V, of the quantum number ,, one assigns the number n, of particles which are in this state. We assume that there exist eigenvectors @n, for every n, and a Hilbert space H which is the closure of the linear aggregate of {,1 n,=0, 1, 2. .}. F. Corresponding to a state # of the field, there exist vectors of the infinite direct product //(R) H, of the space H, in the sense of J. Von Neumann 2. We leave details of the indicate direct product to the original author’s article 2J, and indicate only the following properties (1), (2) of c-sequences which has close relation to the am-

On the ambiguity of cut-off process in the theory of quantum field